Question 128
Given an acute triangle $PBC$, $PB \neq PC$. Let points $A$, $D$ be on sides $PB$ and $PC$, respectively. Let $M$, $N$ be the midpoints of segments $BC$ and $AD$, respectively. Lines $AC$ and $BD$ intersect at point $O$. Draw $OE \perp AB$ at point $E$ and $OF \perp CD$ at point $F$.
(1) Prove that if $A$, $B$, $C$, $D$ are concyclic, then
$$
EM \times FN = EN \times FM.
$$
(2) Are the four points $A$, $B$, $C$, $D$ always concyclic if $EM \times FN = EN \times FM$? Prove your answer.
(1) Prove that if $A$, $B$, $C$, $D$ are concyclic, then
$$
EM \times FN = EN \times FM.
$$
(2) Are the four points $A$, $B$, $C$, $D$ always concyclic if $EM \times FN = EN \times FM$? Prove your answer.